Annuloplasty rings for use as heart valve prostheses are well known as shown, for example, by Alfieri et al. U.S. patent application publication U.S. 2002/0173844 A1 and Bolling et al. U.S. patent application publication U.S. 2003/0093148 A1. Most such rings are substantially planar (see, for example, the above-mentioned Alfieri et al. reference). Recently, an interest in non-planar (e.g., saddle-shaped) rings has developed (see, for example, the above-mentioned Bolling et al. reference). The known non-planar rings tend to be substantially uniformly rigid. It would be desirable to have non-planar rings that are not uniformly rigid.